12.4 Tangent Vectors and Normal Vectors By Dr. Julia Arnold

I am going to list the important theorems and definitions from this section and then we will look at some problems. Definition of the Unit Tangent Vector Let C be a smooth curve represented by r on an open interval I. The unit tangent vector T(t) at t is defined as Definition of Principal Unit Normal Vector Let C be a smooth curve represented by r on an open interval I. If then the Principal Unit Normal Vector at is defined as

Theorem 12.4 Acceleration Vector If r(t) is the position vector for a smooth curve C and N(t) exists , then the acceleration vector a(t) lies in the plane determined by T(t) and N(t). Theorem 12.5 Tangential and Normal Components of Acceleration If r(t) is the position vector for a smooth curve C for which N(t) exists, then the tangential and normal components of acceleration are as follows:

Example 1 find the unit tangent vector to the curve at the specified value of the parameter.

Problem #16 in the HW text.

Problem 22: Verify that the space curves intersect at the given values of the parameters. Find the angles between the tangent vectors to the curve at the point of intersection. Solution:

Problem 28: Find the principal unit normal vector to the curve at the specified value of the parameter. Solution: Continued on next slide

Find v(t), a(t), T(t) and N(t) for r(t)= t2j + k Problem 34 Find v(t), a(t), T(t) and N(t) for r(t)= t2j + k Is the speed changing or unchanging? Solution: V(t) would be the velocity vector and equal to r’(t) a(t) is the acceleration vector and would be the derivative or the velocity vector or r”(t) The speed is variable

Problem 58: Find T(t), N(t), aT, aN, at the given time t for the space curve r(t). [Hint: Find a(t), T(t), aT, and aN. Solve for N in the equation a(t)= aTT + aNN.] Solution:

Problem 58 Continued From slide 2 Continued

a(t)= aTT + aNN.] Problem 58 Continued This is an easier way of finding N rather than differentiating

Definition of Unit Binormal vector: Find the vectors T and N Definition of Unit Binormal vector: Find the vectors T and N. For the vector valued function r( t ) at the given value of t. The Unit Binormal vector is Problem 76 Solution: Continued

This concludes the sample problems from 12.4